Statistical Analysis of Viral Growth Curves

Introduction

This tutorial describes in detail how to implement the analytical methods
described in "A statistical method for comparing viral growth curves" by Wang
and Bushman. It would be helpful to read this paper before going through the
tutorial. In addition, it would be useful to read through some of the introductory
material in GraphPad Prism.

Preliminary considerations

Before beginning the experiment, it is helpful to consider how to maximize the
sensitivity of the final statistical tests. The recommended method involves
generating a summary statistic for each culture--the growth rate--and then
comparing the collection of growth rates among drug concentrations, mutants,
etc. The statistical power is governed by the number of growth rate
measurements in the comparison. For this reason, maximizing the sensitivity
involves carrying out the maximum reasonable number of replicates. If replicates
are expensive, it is wise to carry out the minimal number of measurements per
independent culture, and analyze the maximum feasible number of cultures. It
may be helpful to carry out a pilot study to find the range of exponential growth,
then analyze only this part of the curve in a more extensive series of replicates.
In the analysis below we have sought to keep the methods as simple as possible.
The data is first transformed in Excel, which will be familiar to most virologists.
We then analyze the data using GraphPad Prism
(www.graphpad.com). This
statistical package has the advantage of resembling Excel in its format for data
tables and general character. Other statistical packages could of course be used,
but many of these have much steeper learning curves.

Step1: Data Preparation

Data was tabulated in Excel as shown. Data points below the level of
detection were given a value of 1. Note that designations such as ND
for "not detected" will cause problems in the downstream analysis.
Similarly, zeros are a problem since the logarithm is undefined.

Data was then smoothed by multiplying data from days 5-8 by 4, days 9-12
by 16, days 13-16 by 64, and days 17-20 by 256, to account for splitting
of cells during the culture period. This yielded the data matrix shown.

The natural log was then taken for each data point, yielding the matrix shown.

Data for the tests of integrase inhibitors described in this study can be found here:
Sample data

Step2: Data Preparation

After these transformations, the data was taken into GraphPad Prism.
First, the nature of the data table needs to be specified as shown. Data
from each culture dish is treated as a separate column in the data table.
Choose New Project under the File Menu, and configure "Y columns" with
"A single column of values". Press OK.

The table from Excel is then pasted into the configured GraphPad
data table as shown.

The exponential phase is then selected for study. The times that we plan
to exclude are removed using the Exclude Values function in the Edit
menu. In this example, we block out days 1-5 and 15-20 and analyze only
days 6-14. Note that this choice was made by inspecting graphs of the
time versus log (p24) values, averaged over each treatment, as shown at
left. These graphs can be generated in either Excel or GraphPad Prism.

The edited data table looks like this.

Step3: Data Analysis

Under the Analyze menu, choose the Nonlinear regression option under the Curves and Regression menu.

It is necessary to configure the output by turning off the CI of variables
option. (Otherwise, the numbers are rendered with confidence intervals,
and cannot be further manipulated.)

Then under Equation choose Polynomial: First Order.

Step4: Data Analysis

This yields the linear regression fits for each experimental replicate,
indicated as SLOPE in the table.

We now need to construct a second data table containing the growth rates.
For the statistical analysis, we need to make the four rates for each
replicate a separate column. Thus we need four columns, one for each
condition, each containing four values. Choose New Project under the File
Menu, and configure "Y columns" with "A single column of values". Press OK.

To fill in the new data table, copy the first four SLOPE values, which
correspond to the no-inhibitor control. Then bring up the new table, and
click on the box in the upper left of the the Y-value field (cell A1). We
now want to paste the row of four SLOPE values into a single column with
four entries (i. e. transpose the values). Go to the Edit menu and select
Paste Special. In the Format Data Object dialog box, select Placement,
and in the new dialog box select "Transpose". Then press the "Paste"
button in the bottom right corner. This places the four rates in a single
column. Enter the rates for the three other conditions in the next three
columns.

Step5: Data Analysis

The completed data table looks like this.

We now want to run the non-parametric comparisons of means using the
Kruskal-Wallis test. Go to the analyze menu and select One-way ANOVA
(and nonparametric) under Statistical Analysis. Press OK.

In the next window, check Nonparametric test, which gives you the
Kruskal-Wallis test.

Step6: Data Analysis

We now must set up the control for multiple comparisons. Under Post
Test, select Dunn's: Compare selected pairs of columns. Press Select.

We are now presented with a dialog box allowing us to choose the
comparisons to use.

Use the tool at the top to specify comparison of A to B, A to C, and
A to D. Press OK.

This yields our final answer. The overall P value (line 4) is significant at
P=0.0041, so we have a basis for asking where the differences are present in
the data. Under Dunn's Multiple Comparison Test, we can see that the
no-inhibitor value is significantly different from the 20 uM value, but the
other two concentrations (100 nM and 2 uM) were not significantly different.
If we wished to try to establish that there was a significant difference
between the no-inhibitor values and the 2 uM values, we would need to carry
out another experiment with more independent rate measurements to increase
the statistical power.